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A characteristic class\(\kappa\) is a natural transformation that
associates to each vector bundle \(E \to M\) a cohomology class
\(\kappa(E) \in H^*(M;R)\) such that for any continuous map \(f\colon N \to M\)
from another topological manifold \(N\), the naturality condition is
satisfied:

One way to obtain and compute characteristic classes in the de Rham cohomology
with coefficients in the ring \(\CC\) is via the so-called Chern-Weil theory
using the curvature of a differentiable vector bundle.

For that let \(\nabla\) be a connection on \(E\), \(e\) a local frame on
\(E\) and \(\Omega\) be the corresponding curvature matrix
(see: curvature_form()).

is well-defined, independent of the choice of \(\nabla\) (the proof can be
found in [Roe1988] pp. 31) and fulfills the naturality condition.
This is the foundation of the Chern-Weil theory and therefore the following
definitions.

Note

This documentation is rich of examples, but sparse in explanations. Please
consult the references for more details.

To apply the Chern-Weil approach, we need a bundle connection in terms of a
connection one form. To achieve this, we take the connection induced from the
hermitian metric on the trivial bundle
\(\CC^2 \times \CC\mathbf{P}^1 \supset \gamma^1\). In this the frame \(e\)
corresponds to the section \([z:1] \mapsto (z,1)\) and its magnitude-squared
is given by \(1+|z|^2\):

Usually, there is no such thing as “Pfaffian classes” in literature. However,
using the matrix’ Pfaffian and inspired by the aforementioned definitions,
such classes can be defined as follows.

Let \(E\) be a real vector bundle of rank \(2n\) and \(f\) an odd real function
being analytic at zero. Furthermore, let \(\Omega\) be skew-symmetric, which
certainly will be true if \(\nabla\) is metric and \(e\) is orthonormal. Then
we call

Return the form representing self with respect to the given
connection connection.

INPUT:

connection – connection to which the form should be associated to;
this can be either a bundle connection as an instance of
BundleConnection
or, in case of the tensor bundle, an affine connection as an instance
of AffineConnection

cmatrices – (default: None) a dictionary of curvature
matrices with local frames as keys and curvature matrices as items; if
None, Sage tries to get the curvature matrices from the connection

OUTPUT:

mixed form as an instance of
MixedForm
representing the total characteristic class

Note

Be aware that depending on the characteristic class and complexity
of the manifold, computation times may vary a lot. In addition, if
not done before, the curvature form is computed from the connection,
here. If this behaviour is not wanted and the curvature form is
already known, please use the argument cmatrices.